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MathGroup Archive 1999

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Re: circumference of an ellipse

  • To: mathgroup at
  • Subject: [mg19342] Re: circumference of an ellipse
  • From: "Alan W.Hopper" <awhopper at>
  • Date: Fri, 20 Aug 1999 23:09:28 -0400
  • Sender: owner-wri-mathgroup at


Here is another approach to approximating the ellipse circumference
or perimeter, utilising a 'Complete Elliptic Integral of the 2 nd Kind'
which is incorporated in the Mathematica 3 function - EllipticE.

In[1]:= 4*a*Integrate[Sqrt[1 - e^2*Sin[Theta]^2],{Theta, 0, Pi/2}]

Out[2]= 4 a EllipticE[e^2]

In[3]:= a = 1;  e = 0; 4 a EllipticE[e^2]

Out[4]= 2 Pi

Where, (e) Eccentricity = Sqrt[1-(b/a)^2]

In[5]:= ellipsePerimeter[a_,b_,n_]:= Module[{e = Sqrt[1-(b/a)^2]},
	        	N[4 a EllipticE[e^2], n]]

In[6]:= ellipsePerimeter[2,1,10]

Out[7]= 9.688448221

In[8]:= ellipsePerimeter[100, 20, 30]

Out[9]= 420.20089079378001889398329177

It is interesting that in contrast to the above that there is a
very simple exact formula for the ellipse area ;  A = Pi a b . 

Alan W. Hopper

awhopper at

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